An Anti-Ramsey Theorem

Let be the Turán number which gives the maximum size of a graph of order containing no subgraph isomorphic to .

In 1973, Erdős, Simonovits and Sós [5] proved the existence of an integer such that for every integer , the minimum number of colours , such that every -colouring of the edges of which uses all the colours produces at least one all whose edges have different colours, is given by . However, no estimation of was given in [5]. In this paper we prove that for . This formula covers all the relevant values of n and p.